The British soldiers are seen to have Lee Enfield rifles and Webley pistols. The Lee Enfield was not adopted by the British Army until 1896, six years after the setting of the film. The Webley model seen was not adopted until the First World War, although a different one was in use at this time.
When Frank is telling Jazira about his Native American background, he says that "shunka wankan" is Sioux for "big dog". The people of this tribe prefer to call themselves "Dakota" or "Lakota", as "sioux" is a word meaning "snake" and was given to the Lakotas by the Crows (enemies of the Lakotas).
During the course of the movie, the pattern on the right side of Hidalgo's neck changes. At his shoulder, throughout the movie, the pattern changes to having chestnut spots and then having a solid white splotch on his shoulder.
When Frank is in the sheik's tent for the first time, and the sheik's horse goes through the tent, it's night outside. When Frank leaves, the sheik wishes him a good night, but you can see the sunlight coming in through the tent flap the horse just went through.
One of the purebred Arabian horses is actually an Andalusian. Although it's not totally implausible that such a horse would be there, claiming it is an Arabian purebred is akin to claiming that the Chimichanga originated in Great Britain.
While the train is pulling out prior to Hopkins getting off it shows the rail cars using a knuckle coupler. Although this coupler was not required by law in the US until 1893, 3 years after the movie takes place, it was invented in 1863 and used by some railroads before legally required.
When Frank goes to shake the Sheik's hand, he is told the Sheik can't touch him or he will lose his ability to foretell the future. But just minutes later, when Frank hands the Sheik the Colt revolver, and the Sheik returns it to him, their hands touch.
When Hidalgo collapses just before Frank has his visions he has black markings in a horseshoe pattern under his nostrils. In shots of Hidalgo's head during the sprint scene that follows, the horse is pure white under his nostrils. In the subsequent victory scene in the ocean, Hidalgo once again has the black markings.